Computational Materials Science 50 (2011) 2663–2665
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Computational Materials Science
journal homepage: www.elsevier.com/locate/commatsci
Self-diffusion in Zn4Sb3 from first-principles molecular dynamics
Ole Martin Løvvik a,b,⇑, Protima Rauwel a, Øystein Prytz a
a
b
University of Oslo, Department of Physics, P.O. Box 1048 Blindern, NO-0316 Oslo, Norway
SINTEF Materials and Chemistry, Forskningsveien 1, 0314 Oslo, Norway
a r t i c l e
i n f o
Article history:
Received 4 March 2011
Received in revised form 4 April 2011
Accepted 9 April 2011
Available online 4 May 2011
a b s t r a c t
The Zn4Sb3 system is promising for thermoelectric applications due to its very low thermal conductivity
coupled with a good power factor. Molecular dynamics calculations based on density functional theory
were carried out for different stoichiometries of Zn4Sb3, corresponding to three situations: the composition Zn3.6Sb3 (actually Zn6Sb5 with only one Zn site occupied), a slightly higher Zn content Zn3.8Sb3 (with
some of the Zn atoms in interstitial sites), and a slightly lower Zn content Zn3.4Sb3 (with some Zn vacancies). The diffusivity was calculated for different temperatures and the diffusion coefficient plotted in
Arrhenius plots. The results compare well with experimental data, and point to a highly mobile Zn species
with a very high diffusion coefficient prefactor.
Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction
Zinc antimonide Zn4Sb3 is one of the more promising materials
for thermoelectricity, partly due to a very low thermal conductivity; this has been linked to the existence of several interstitial Zn
sites [1]. The presence of nano-inclusions and voids has also been
suggested as a mechanism for reducing thermal conductivity,
based on recent high-resolution transmission electron microscope
studies [2–4]. The formation of these nanostructures can be understood from a temperature-dependent solubility of Zn on interstitial
sites in the Zn4Sb3 phase, combined with the diffusivity of Zn being
higher than that of Sb [3,4]. It is already known from a tracer diffusivity experiment that Zn is highly mobile in Zn4Sb3 [5], but it
is not known how the diffusivity of Sb compares to that of Zn.
We have in the present study used first-principles molecular
dynamics (FPMD) to quantify the diffusion coefficient of both Zn
and Sb as a function of temperature in Zn4Sb3.
2. Methodology
Density functional theory (DFT) at the PBE-GGA level [6] was
used to provide self-consistent electronic structures as source for
the calculations of Hellman–Feynman forces. Plane-waves were
used as basis functions according to the projector augmented wave
method [7], as implemented in the Vienna ab initio Simulation
Package (vasp) [8,9]. The cut-off energy of the plane wave expansion was 208 eV, and only the C point was used for the k-space
integrals. The criterion for electronic convergence was a change
in calculated total energy of less than 1 meV between two consec⇑ Corresponding author at: University of Oslo, Department of Physics, P.O.
Box 1048, Blindern, NO-0316 Oslo, Norway. Tel.: +47 93015091.
E-mail address: o.m.lovvik@fys.uio.no (O.M. Løvvik).
0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.commatsci.2011.04.015
utive iterations. This level of precision was sufficient to achieve errors in the calculated forces <0.05 eV/Å, thus giving reliable MD
results. Three different models were employed in order to compare
situations with varying Zn content. The simplest structure had only
the crystallographic site Zn1 fully occupied, with the stoichiometry
Zn3.6Sb3. An alternative model was created by removing Zn atoms,
thus forming Zn vacancies (Zn3.4Sb3), while the last model had Zn
interstitial sites occupied (Zn3.8Sb3). The standard R-3c unit cell
was used, which consisted of 64–68 atoms. A Boltzmann distribution of velocities corresponding to the given temperature was
automatically provided by vasp. The time step used for molecular
dynamics (MD) was 1 fs, and the simulations were run for 5000
time steps.
3. Results and discussion
The mean square displacement (MSD) of Zn and Sb in Zn4Sb3 is
shown in Fig. 1. It is evident that it generally takes 2–3 ps before
the MSD starts growing linearly for all the compositions. This linear growth is a sign of solid state diffusion, confirmed by inspection of the ionic movements. Also, most of the temperatures of
this study were below the melting point of Zn4Sb3 (836 K [10]).
Furthermore, melting did not occur at the highest temperatures;
this was confirmed by assessing the total electronic energy as a
function of time and temperature (not shown here). Such overheating is quite common in molecular dynamics, and is in our case convenient – it gives access to a larger range of temperatures than
would otherwise be accessible (i.e. only below the melting point).
Another signature of solid state diffusion can be seen in a plot of
the diffusion coefficient D as a function of time (Fig. 2). After 1–4 ps
the diffusion coefficient takes on a more or less constant value. This
value (calculated as the average of D during the last ps of the
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O.M. Løvvik et al. / Computational Materials Science 50 (2011) 2663–2665
Fig. 1. The mean square displacement (MSD) as a function of time for Sb diffusion (left) and Zn diffusion (right) in Zn4Sb3. Results for the compositions Zn3.4Sb3, Zn3.6Sb3 and
Zn3.8Sb3 are shown as dotted, solid, and dashed curves, respectively. The temperature was 700 K.
simulation time) was used to create Arrhenius plots of D as a
function of the temperature.
This is shown in Fig. 3 for the three different Zn4Sb3 compositions. Despite the rather large uncertainty of the methodology,
we see that a linear fit to the logarithm of D versus reciprocal temperature is reasonable. The linear fits in Fig. 3 were used to estimate the activation energy Q, which is the slope of the curve.
Furthermore, the diffusion coefficient prefactor D0, which is the
point of interception between the trend line and the ordinate axis,
was obtained by extrapolating the linear trend line as shown in
Fig. 3.
The results are summarized in Table 1, where D0 and Q are given
for Zn and Sb for all the compositions. They are also compared to
experimental results from Ref. [5]. We first note that the quality
of the linear fit (quantified by R2) is relatively high for most of
the curves, except that of Sb diffusion in Zn3.4Sb3 – we will therefore neglect these results in the following discussion. It is then
interesting to see that the activation energy is quite similar for
Zn and Sb for all three compositions; between 11 and 15 kJ/mol.
This compares very favorably with the experimental result for Q,
which is 11 kJ/mol [5]. This means that the activation energy is
quite accurately predicted with this technique, and that we can
quite safely conclude that Q is similar for Sb and Zn. We can also
see that Q consistently increases with increasing Zn content. This
is probably due to local distortions of the lattice close to the interstitial Zn atoms, effectively increasing the diffusion barrier.
However, when we turn to the diffusion coefficient prefactor D0,
the picture is different. First of all, D0 for Zn is approximately twice
as large as that for Sb, leading to a very high overall mobility of Zn.
But we also note that the calculated prefactors are two orders of
magnitude higher than the experimental one. This is not unusual
for calculated prefactors [11,12], and can be rationalized by the relatively short simulation times being used by MD calculations. The
prefactor is determined by jump frequencies and correlation factors; [13] the latter designate the deviation from true random walk
due to correlated jumps. As an example, a jump from a lattice site
to an interstitial site can be strongly correlated with a jump back to
the lattice site, which would lead to a correlation factor considerably smaller than unity. The jump frequencies and correlation factors may have significantly different time signatures. The jump
frequencies are undoubtedly well described by the MD calculations, where several jumps are seen within the simulation times.
However, there are most probably correlation effects which hinder
long-distance diffusion; such effects can have significantly longer
time constants than the jump frequencies, and may thus not be
seen until considerably longer simulation times are employed.
When comparing our study with the experiments, we should note
that we have studied self-diffusion, while Ref. [5] used diffusion of
a tracer species to quantify the diffusivity – this is a possible additional contribution to the discrepancy between the theoretical and
experimental results.
The question is then whether we can trust the relative size of
the calculated prefactors for Sb and Zn – is really D0 larger for
Zn, or can the correlation factors change the relative mobility of
the two species? It is difficult to give a definite answer to this question, but one can speculate that the various interstitial Zn sites can
provide more opportunities for the Zn atoms to escape through
non-correlated jumps. This can possibly change the correlation factor in favor of Zn, increasing rather than decreasing the difference
between Zn and Sb. This remains of course speculative, and only
Fig. 2. The diffusion coefficient D as a function of time for Sb diffusion (left) and Zn diffusion (right) in Zn4Sb3. Results for the compositions Zn3.4Sb3, Zn3.6Sb3 and Zn3.8Sb3 are
shown as dotted, solid, and dashed curves, respectively. The temperature was 700 K.
O.M. Løvvik et al. / Computational Materials Science 50 (2011) 2663–2665
2665
Fig. 3. The logarithm of D as a function of reciprocal temperature for Sb diffusion (left) and Zn diffusion (right) in Zn4Sb3. Linear trends for the compositions Zn3.4Sb3, Zn3.6Sb3
and Zn3.8Sb3 are shown as dotted, solid, and dashed lines, respectively. The temperature was varied between 400 and 1000 K.
Table 1
The Diffusion Coefficient Prefactor D0 and the activation energy Q for the Different
Compounds in this Study, compared with experimental results for Zn diffusion in
Zn4Sb3 [5]. The quality of the linear fit (the straight lines in Fig. 3) is measured by the
coefficient of determination R2.
4
cm2/s)
Q (kJ/mol)
R2 (1)
Compound
D0 (10
Sb
Zn
Sb
Zn
Sb
Zn
Zn3.4Sb3
Zn3.6Sb3
Zn3.8Sb3
Expt. [5]
0.82
1.3
1.8
2.8
2.6
3.4
0.13
7
13
14
11
13
15
11
0.79
0.93
0.85
0.96
0.96
0.98
experiments or a thorough calculation of the correlation factors
can resolve this matter properly.
4. Conclusions
We have shown that Zn has a very high mobility in Zn4Sb3,
based on first-principles molecular dynamics calculations. The
activation energies of Zn and Sb jumps are quite similar, and close
to the experimental value. A very high prefactor D0 makes Zn diffusion significantly faster than Sb diffusion, but this is quite speculative, since the experimental value of D0 for Zn is two orders of
magnitude smaller than the calculated one. We nevertheless believe that the results support the assumptions of recent articles
[3,4], where it was proposed that rapid diffusion of Zn can lead
to Zn precipitates and subsequently nanovoids.
Acknowledgment
The Research Council of Norway is gratefully acknowledged for
financial support, and the NOTUR consortium for a grant of computational time. Ole Bjørn Karlsen, Espen Flage-Larsen and Espen
Sagvolden are acknowledged for stimulating discussions.
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